A268309 -id:A268309 - cp's OEIS frontend

A138177 Triangle T(n,k) read by rows: number of k X k symmetric matrices with nonnegative integer entries and without zero rows or columns such that sum of all entries is equal to n, n>=1, 1<=k<=n.

1, 1, 2, 1, 4, 4, 1, 7, 15, 10, 1, 10, 36, 52, 26, 1, 14, 74, 176, 190, 76, 1, 18, 132, 460, 810, 696, 232, 1, 23, 222, 1060, 2705, 3756, 2674, 764, 1, 28, 347, 2180, 7565, 15106, 17262, 10480, 2620, 1, 34, 525, 4204, 19013, 51162, 83440, 80816, 42732, 9496, 1, 40

Offset: 1

Vladeta Jovovic, Mar 03 2008

See the Brualdi/Ma reference for the connection to A161126. - Joerg Arndt, Nov 02 2014
T(n,k) is also the number of semistandard Young tableaux of size n whose entries span the interval 1..k. See also Gus Wiseman's comment in A138178. The T(4,2) = 7 semi-standard Young tableaux of size 4 spanning the interval 1..2 are:
   11  122  112  111  1222  1122  1112
   22  2    2    2                      . - Jacob Post, Jun 15 2018

			Triangle T(n,k) begins:
  1;
  1,  2;
  1,  4,   4;
  1,  7,  15,   10;
  1, 10,  36,   52,   26;
  1, 14,  74,  176,  190,   76;
  1, 18, 132,  460,  810,  696,  232;
  1, 23, 222, 1060, 2705, 3756, 2674, 764;
  ...

Alois P. Heinz, Rows n = 1..141, flattened
Richard A. Brualdi, Shi-Mei Ma, Enumeration of involutions by descents and symmetric matrices, European Journal of Combinatorics, vol.43, pp.220-228, (January 2015).
FindStat - Combinatorial Statistic Finder, Semistandard Young tableaux
Samantha Dahlberg, Combinatorial Proofs of Identities Involving Symmetric Matrices, arXiv:1410.7356 [math.CO], (27-October-2014)

Cf. (row sums) A138178, A135589, A135588, A161126, A210391.
Main diagonal gives A000085. - Alois P. Heinz, Apr 06 2015
T(2n,n) gives A266305.
T(n^2,n) gives A268309.

gf:= k-> 1/((1-x)^k*(1-x^2)^(k*(k-1)/2)):
A:= (n, k)-> coeff(series(gf(k), x, n+1), x, n):
T:= (n, k)-> add(A(n, k-i)*(-1)^i*binomial(k, i), i=0..k):
seq(seq(T(n, k), k=1..n), n=1..12);  # Alois P. Heinz, Apr 06 2015

gf[k_] := 1/((1-x)^k*(1-x^2)^(k*(k-1)/2)); A[n_, k_] := Coefficient[ Series [gf[k], {x, 0, n+1}], x, n]; T[n_, k_] := Sum[(-1)^j*Binomial[k, j]*A[n, k-j], {j, 0, k}]; Table[T[n, k], {n, 1, 12}, {k, 1, n}] // Flatten (* Jean-François Alcover, Jan 31 2016, after Alois P. Heinz *)

T(n,k) = Sum_{i=0..k} (-1)^i * binomial(k,i) * A210391(n,k-i). - Alois P. Heinz, Apr 06 2015

A266305 Number of n X n symmetric matrices with nonnegative integer entries and without zero rows or columns such that the sum of all entries is equal to 2n.

Original entry on oeis.org

1, 1, 7, 74, 1060, 19013, 408650, 10219360, 291158230, 9302358947, 329192040880, 12775809098058, 539351216354728, 24600280965461923, 1205263251360664310, 63115789721408960624, 3517483455875467926588, 207834769804597591153769, 12976002600530598793672490

Offset: 0

Alois P. Heinz, Jan 31 2016

nonn

			a(2) = 7:
  [1 1]  [2 1]  [0 1]  [2 0]  [0 2]  [3 0]  [1 0]
  [1 1]  [1 0]  [1 2]  [0 2]  [2 0]  [0 1]  [0 3].

Alois P. Heinz, Table of n, a(n) for n = 0..200

Cf. A138177, A268309.

gf:= k-> 1/((1-x)^k*(1-x^2)^(k*(k-1)/2)):
A:= (n, k)-> coeff(series(gf(k), x, n+1), x, n):
a:= n-> add(A(2*n, n-j)*(-1)^j*binomial(n, j), j=0..n):
seq(a(n), n=0..20);

gf[k_] := 1/((1-x)^k*(1-x^2)^(k*(k-1)/2)); A[n_, k_] := SeriesCoefficient[ gf[k], {x, 0, n}]; a[n_] := Sum[A[2*n, n-j]*(-1)^j*Binomial[n, j], {j, 0, n}]; Table[a[n], {n, 0, 20}] (* Jean-François Alcover, Feb 25 2017, translated from Maple *)

a(n) = A138177(2n,n).

cp's OEIS Frontend

A138177 Triangle T(n,k) read by rows: number of k X k symmetric matrices with nonnegative integer entries and without zero rows or columns such that sum of all entries is equal to n, n>=1, 1<=k<=n.

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